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Properties

Label 2-605-11.4-c1-0-22
Degree $2$
Conductor $605$
Sign $0.970 - 0.242i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.09 + 0.796i)2-s + (0.177 − 0.547i)3-s + (−0.0501 − 0.154i)4-s + (0.809 − 0.587i)5-s + (0.631 − 0.458i)6-s + (1.12 + 3.47i)7-s + (0.905 − 2.78i)8-s + (2.15 + 1.56i)9-s + 1.35·10-s − 0.0933·12-s + (−2.29 − 1.66i)13-s + (−1.52 + 4.70i)14-s + (−0.177 − 0.547i)15-s + (2.95 − 2.14i)16-s + (2.98 − 2.17i)17-s + (1.11 + 3.44i)18-s + ⋯
L(s)  = 1  + (0.775 + 0.563i)2-s + (0.102 − 0.315i)3-s + (−0.0250 − 0.0771i)4-s + (0.361 − 0.262i)5-s + (0.257 − 0.187i)6-s + (0.426 + 1.31i)7-s + (0.320 − 0.985i)8-s + (0.719 + 0.522i)9-s + 0.428·10-s − 0.0269·12-s + (−0.635 − 0.461i)13-s + (−0.408 + 1.25i)14-s + (−0.0459 − 0.141i)15-s + (0.738 − 0.536i)16-s + (0.724 − 0.526i)17-s + (0.263 + 0.811i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53250 + 0.311312i\)
\(L(\frac12)\) \(\approx\) \(2.53250 + 0.311312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-1.09 - 0.796i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.177 + 0.547i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.12 - 3.47i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.29 + 1.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.98 + 2.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0293 + 0.0904i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + (-2.08 - 6.42i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.48 + 3.98i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.04 - 9.35i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.57 + 7.91i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 + (0.687 - 2.11i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.42 + 1.75i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.62 + 8.09i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.86 - 4.98i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (6.71 - 4.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.407 - 1.25i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.2 + 8.15i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.61 - 6.25i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-3.50 - 2.54i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.59115441216198153057845301370, −9.757520670079980577633739972416, −8.942219747841255649506526372844, −7.80898091548775440959500955985, −7.03904491545502420892169484670, −5.87114818973755173356005597198, −5.25740624906330503541276577750, −4.53763806786535294748390340695, −2.86649346233235674369881448385, −1.51579959284313294205641896248, 1.55727474106292716196062021165, 3.04287723742262772834507340079, 4.10859035886999789234462141083, 4.54810827931948857519636946060, 5.87093311506560080818596244313, 7.18841215272628241479336145181, 7.74998818043972677221065459409, 9.064456989534793561494320901265, 10.03451103688859295204022683557, 10.65948712126384385241084697113

Graph of the $Z$-function along the critical line